Building on Sir Sam’s Formalism: Molecularly-Informed Field-Theoretic

Simulations of Soft Matter

Glenn H. Fredrickson Departments of Chemical Engineering & Materials

Materials Research Laboratory (MRL)

University of California, Santa Barbara

R&D Strategy Office Mitsubishi Chemical Holdings Corporation

Tokyo, Japan

TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAA

Acknowledgements Postdocs:

Nabil Laachi Xingkun Man Rob Riggleman Scott Sides, Eric Cochran Yuri Popov, Jay Lee Venkat Ganesan

Students: Mike Villet Erin Lennon Su-Mi Hur Debbie Audus

Collaborators: Kris Delaney (UCSB)

Henri Orland (Saclay) Hector Ceniceros (UCSB) Carlos Garcia-Cervera (UCSB)

Funding: NSF DMR-CMMT NSF DMR-MRSEC US Army (ICB)

Complex Fluids Design Consortium (CFDC):

Rhodia Mitsubishi Chemical Arkema Dow Chemical Nestlé Kraton Polymers DSM Intel, JSR, Asahi Kasei, Samsung IBM, SK Hynix SNL, LANL, ARL

Sam’s Favorite Complex Gaussian Integrals

Representing:

Pair interaction (Re v > 0)

Inverse operator (Re L > 0)

True for scalars, vectors, and functions w and !

Outline

Field-theoretic simulations Why?

Methodology

Applications Advanced lithography – directed self-

assembly

Polyelectrolyte complexation

Fun Superfluid He

Why nano-structured polymers?

Nano-structuring is a way to achieve functionality that differentiates and adds value to existing and new families of polymers and derivative materials

We aim to develop simulation tools that can guide the design of nano/meso-structured polymer formulations and

soft materials

• Microphase separation of block copolymers

Nanoscale Morphology Control: Block Copolymers

B

S

S S Holden & Legge (Shell – Kraton Polymers)

SBS Triblock Thermoplastic Elastomer

f

10 nm

Elastic, clear Rigid, tough, clear

Why Field-Based Simulations?

Relevant spatial and time scales challenging for fully atomistic, “particle-based” simulations

Use of fluctuating fields, rather than particle coordinates, has computational advantages:

Simulations become easier at high density & high MW – access to a mean-field (SCFT) solution

Systematic coarse-graining more straightforward

ABA + A alloy, S. W. Sides

Nano/meso: 1 nm to 1 μm

3x3x3 unit cells of Fddd (O70) phase in ABC triblock, K. Delaney

2.5 µm

Models

Starting point is a coarse-grained particle model

Continuous or discrete chain models

Pairwise contact interactions Excluded volume v, Flory

parameters

Easily added: Electrostatic interactions

Incompressibility (melt)

Arbitrary branched architectures

A branched “multiblock” polymer

Sam’s pseudo-potential

Sam’s First Integral: Auxiliary Field Formalism

Representing:

Pair interaction (Re v > 0)

True for scalars, vectors, and functions w!

: an “auxiliary field”

From Particles to Fields

A “Hubbard-Stratonovich-Edwards” transformation is used to convert the many-body problem into a statistical field theory

Boltzmann weight is a complex number!

Polymers decoupled!

microscopic particle density

Edwards Auxiliary Field (AF) Model

Sam’s classic model of flexible homopolymers dissolved in good, implicit solvent (S. F. Edwards, 1965)

Field-theoretic form

Q[iw] is the single-chain partition function for a polymer in an imaginary potential field iw

“Effective Hamiltonian”

Single-Chain Conformations

Q[iw] calculated from propagator q(r,s) for chain end probability distribution

Propagator obtained by integrating a complex diffusion (Fokker-Planck) equation along chain contour s

s

Numerically limiting “inner loop” in field-based simulations!

Observables and Operators

0

s

q(r,s) N

r

q (r,N-s)

•Observables can be expressed as averages of operators O[w] with complex weight exp(-H[w])

•Density and stress operators (complex) can be composed from solutions of the Fokker-Planck equation

Types of Field-Based Simulations

• The theory can be simplified to a “mean-field (SCFT)” description by a saddle point approximation:

• SCFT is accurate for

• We can simulate a field theory at two levels:

dense, high MW melts

“Mean-field” approximation (SCFT): F H[w*]

Full stochastic sampling of the complex field theory: “Field-theoretic simulations” (FTS)

High-Resolution SCFT/FTS Simulations

By spectral collocation methods and FFTs we can resolve fields with > 107 basis functions

Unit cell calculations for ordered phases

Large cell calculations for exploring self-assembly in new systems: “discovery mode”

Flexible code base (K. Delaney) NVIDIA GPUs, MPI, or OpenMP

Confined BC films

2.5 µm

Triply-periodic gyroid phase of BCs

Block copolymer-homopolymer blend

Complex Architectures

Directed Self-Assembly

An emerging sub 20 nm, low cost patterning technique for electronic device manufacturing

Jeong et al., ACS Nano, 4 5181, 2010

1. Can we understand defect

energetics and kinetics at a

fundamental level?

2. Is the ITRS target of < 0.01

defects/cm2 feasible?

Grapho-epitaxy Chemo-epitaxy Unlike bulk BCP assembly, must manage: • Surface/substrate

interactions • Commensurability

Directed Self-Assembly (DSA) for the “hole shrink” problem

Vertical Interconnect Access (VIA) lithography: Use DSA to produce high-

resolution cylindrical holes with reduced critical dimensions relative to a

cylindrical pre-pattern created with conventional lithography

Current metrology is top-down and cannot probe 3D structures!

SiN / SiON

~50nm

~100nm

~20nm

PS

PM

MA

SEM images courtesy of J. Cheng of IBM Almaden Research Center

Prepattern CD (in units of Rg)

Segregation strength χN

SCFT simulations of PS-b-PMMA diblock copolymer in cylindrical prepattern (fPMMA = 0.3)

Hole depth is ~100 nm

Hole CD is varied between 50 and 75 nm

Search for basic morphologies: PMMA-attractive pre-patterns

7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12

15

20

25

Top view

Side view

N. Laachi, S. Hur

High defect energies for PMMA selective walls in pure DB and blends

20 kT defect formation energy = parts per billion defect levels

PS-b-PMMA, χN = 25, CD ~ 68nm (panel 8)

PS-b-PMMA + PMMA, χN = 25, CD ~ 75nm (panel 11)

Nabil Laachi

“String method” for VIAs: Transition pathways

Metastable defects

48 kT 35 kT 15 kT 0 kT

Perfect state

- Multi-barrier pathway - Barriers for defect melting are < 5kT

PMMA-selective prepattern, CD ≈ 85 nm

Weinan, E., Ren, W. and Vanden-Eijnden, E. J. Chem. Phys., 126, 164103 (2007)

N. Laachi et. al., J. Polym. Sci: Part B, Polym. Phys. 53, 142 (2015)

Beyond Mean-Field Theory: the “Sign Problem”

When sampling a complex field theory, the statistical weight exp( – H[w]) is not positive semi-definite

Phase oscillations associated with exp(- i HI[w]) dramatically slows the convergence of Monte Carlo methods based on the positive weight exp(-HR[w])

This sign problem is encountered in other branches of chemistry and physics: QCD, lattice gauge theory, correlated electrons, quantum rate processes

Complex Langevin Dynamics (G. Parisi, J. Klauder 1983)

A Langevin dynamics in the complex plane for sampling complex field theories and avoiding the sign problem

Thermal noise is asymmetrically placed and is Gaussian and white satisfying a fluctuation-dissipation relation:

The stochastic field equations are stiff, nonlocal, and nonlinear

E. M. Lennon et. al., SIAM Multiscale Modeling and Simulation 6, 1347 (2008) M. Villet and GHF, J. Chem. Phys. 132, 034109 (2010)

Curing UV Divergences

FT models (beyond mean-field) with infinite interactions at contact have no well-defined continuum limit

It is critical to “regularize”, i.e. remove, these singularities for results independent of the computational grid

A simple and universal procedure is due to Z.G. Wang (2010):

In the field theory representation:

Smear particles by a Gaussian of width a

Polyelectrolyte Complexation: Complex Coacervates

• Aqueous mixtures of

polyanions and

polycations complex to

form dense liquid

aggregates – complex

coacervates

• Applications include:

• Food/drug

encapsulation

• Drug/gene delivery

vehicles

• Artificial membranes

• Bio-sensors

• Bio-inspired adhesives

Cooper et al (2005) Curr Opin Coll. & Interf. Sci. 10, 52-78.

+ + +

-

- -

Herb Waite (UCSB) Bioadhesives:

Sand Castle Worms, Marine Mussels

A Symmetric Coacervate Model

• In the simplest case, assume symmetric polyacids & polybases mixed in equal proportions

• Polymers are flexible and carry total charge

• Implicit good solvent

• Interactions: Coulomb and excluded volume

Uniform dielectric medium:

+ + +

-

- -

Corresponding Field-Theory Model

lB =e2 / kBT: Bjerrum length

v: excluded volume parameter

σ: charge density

w: fluctuating chemical pot.

: fluctuating electrostatic pot.

polymer partition

function

Harmonic Analysis

• Three dimensionless parameters appear in the model (and a/Rg ):

• Model has a trivial homogeneous mean-field solution, with no coacervation predicted

• Expanding H to quadratic order in w and we recover the “RPA” result of Castelnovo, Joanny, Erukimovich, Olvera de la Cruz, …

• These attractive electrostatic correlations provide the driving force for complexation However, we can numerically simulate the exact model!

C = 2, B = 1, E = 64000

FTS-CL Simulations of Complex Coacervation

Y. O. Popov, J. Lee, and G. H. Fredrickson,

J. Polym. Sci. B: Polym. Phys. 45, 3223 (2007)

Electrostatic potential fluctuations necessary to obtain coacervation!

Complex Coacervation vs “Self-Coacervation”

Using FTS-CL, we have generated the first “exact” phase diagrams for complexation of blends and polyampholytes (B=1, a/Rg = 0.2)

RPA fails qualitatively on the dilute branches!

vs.

K. Delaney

RPA

Evidence for Dimerization

Many authors have speculated on dimerization in the dilute branch prior to complexation (Rubinstein, Dobrynin, Ermoshkin, Shklovskii, …)

K. Delaney

B=1, C = 1e-3

Polyampholyte and blend show same +- correlations at large E dimerization!

Other Field Theory Representations?

There are at least two alternatives to Sam’s “Auxiliary Field (AF)” representation of polymer field theory that have the nonlocal/nonlinear character of the Hamiltonian more simply expressed

Another framework due to Sam is the “Coherent States (CS)” representation, an approach for branched polymers adapted from Edwards and Freed and inspired by quantum field theory

Our innovation: adapting the CS to linear polymers and the first numerical simulations!

S. F. Edwards, K. F. Freed J. Phys. C: Solid St. Phys. 3, 739 (1970)

Sam’s Second Integral: Coherent States

(Re L > 0)

The kernel of L-1 is a free polymer propagator or “Green’s function”

For

r r' s

CS Representation: Ideal Polymers

In the CS framework, the degrees of freedom are D+1 dimensional forward and backward “propagator-like” fields:

CS representation of grand partition function of an ideal gas of linear polymers:

z: polymer activity Q0: single polymer partition function

L

L-1

K. Delaney, X. Man, H. Orland, GHF

,

Chain end sources

CS Representation: Interacting Polymers

CS representation of Edwards Solution Model in Grand Canonical Ensemble:

The same Edwards model, more simply expressed!

Segment density operator

Complex Langevin for CS Framework

The standard “diagonal” CL scheme is numerically unstable for the CS representation. A stable scheme has an off-diagonal mobility matrix:

The complex noise terms are not uniquely specified by CL

theory, but a suitable choice is

ηi are real, uncorrelated, D+1 dimensional Gaussian, white noises

h́ i ( r ; s; t ) ´ j ( r 0; s0; t0) i = ±i j ±( r ¡ r0) ±( s ¡ s0)

Population standard deviation of the density operator

Current Limitation of CS Framework for Polymers

Ratio of simulation steps needed to produce a specified error of the mean

Current CL sampling of CS models is currently too noisy to

be practically useful…but these are early days!

4He Phase Behavior

4He at low temperature has normal and superfluid liquid phases; the latter a manifestation of collective quantum behavior of cold boson systems

A fraction of the fluid in the superfluid state has zero viscosity!

The superfluid is weird in other ways with angular momentum quantized and thermal fluctuations/agitation spawning vortices (“rotons”)

http://ltl.tkk.fi/research/theory/helium.html

Many cool images of superfluid He fountains and vortices can be found online!

Quantum Statistical Mechanics: Diffusion in Imaginary Time (One Particle, N=1)

Free particle diffusion in imaginary time in a closed cycle with τ = 1:

“density matrix”

“thermal wavelength”

Built from 1-particle eigenstates, energies

R. Feynman

Λ = 10 Å at 1K for He4

Many Particles: Quantum Indistinguishability

The many body wave function for identical Bose particles must be symmetric under pair exchange. For N indistinguishable particles:

• Each permutation of labels P can be decomposed into cycles (“ring polymers”) of imaginary time propagation – these are “exchange interactions”

• Since Λ is large for He4 at low T, these cycles dominate the thermodynamics and are responsible for superfluidity!

Coherent States Formulation of QFT

For interacting bosons in the grand canonical ensemble and periodic BCs on τ to enforce closed “ring polymers”,

• Sums all closed cycles with both exchange and pairwise interactions among bosons!

• Unlike classical polymers, the interactions are at equal τ

pair potential

chemical potential

Ideal gas (non-interacting) 4He atoms

Classical Quantum BEC

Complex Langevin simulations, K. Delaney First direct simulation of a CS Quantum Field Theory!

Interacting 4He system

Perturbation theory for λ line (S. Sachdev, 1999)

Quantum critical point

Contact potential

Normal fluid

Superfluid

K. Delaney

Future Work: Optical Lattices

Our CS-CL simulation framework seems well positioned to tackle current research on cold bosons in periodic potentials It is likely advantaged over path integral quantum Monte Carlo techniques at low T, high ρ

Discussion and Outlook “Field-based” computer simulations are powerful tools for exploring

self-assembly in polymer formulations SCFT well established

Field-Theoretic Simulations (FTS) maturing

Non-equilibrium extensions still primitive

Good numerical methods are essential! Complex Langevin sampling is our main tool for addressing the sign

problem; stable semi-implicit or exponential time differencing schemes mandatory

Variable cell shape and Gibb’s ensemble methods now exist

Free energy accessible by thermo integration; flat histogram?

Coarse-graining/RG techniques improving

Coherent states (CS) formalism looks promising Semi-local character may offer computational advantages, e.g. in

coarse-graining, while still allowing for treatment of a wide range of systems

Currently too noisy, but much remains unexplored

Potential for advancing current research in cold bosons?

G. H. Fredrickson, The Equilibrium Theory of Inhomogeneous Polymers (Oxford 2006)

Framework due to Sam!